Question

According to the Bohr theory of hydrogen atom, the speed of the electron, its energy and the radius of its orbit varies with the principal quantum number n, respectively, as

• A. $\frac{1}{n}, \frac{1}{n^{2}}, n^{2}$
• B. $\frac{1}{n}, n^{2}, \frac{1}{n^{2}}$
• C. $n^{2}, \frac{1}{n^{2}}, n^{2}$
• D. $n, \frac{1}{n^{2}}, \frac{1}{n^{2}}$

Answer 1

Answer: (A)

According to Bohr's theory of hydrogen atom
(i) The speed of electron in $n$th orbit
$v_{n}=\frac{Z e^{2}}{2 \varepsilon_{0} n h}$
or $v_{n} \propto \frac{1}{n}$
(ii) The energy of electron in the $n$th orbit
$E_{n}=-\left(\frac{m e^{4}}{8 \varepsilon_{0}^{2} h^{2}}\right) \cdot \frac{Z^{2}}{n^{2}}$
$=-13.6 \frac{Z^{2}}{n^{2}} eV$
or $E_{n} \propto \frac{1}{n^{2}}$
(iii) The radius of the electron in the $n$th orbit
$r_{n}=\frac{n^{2} h^{2} \varepsilon_{0}}{\pi m Z e^{2}}=0.53 \frac{n^{2}}{2} \mathring{A}$
$r_{n} \propto n^{2}$